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Linear structural causal models (SCMs) are used to express and analyze the relationships between random variables. Direct causal effects are represented as directed edges and confounding factors as bidirected edges. Identifying the causal parameters from correlations between the nodes is an open problem in artificial intelligence. In this paper, we study SCMs whose directed component forms a tree. Van der Zander et al. give a PSPACE-algorithm for the identification problem in this case, which is a significant improvement over the general Gröbner basis approach, which has doubly-exponential time complexity in the number of structural parameters. However, they do not show that their algorithm is complete. In this work, we present a randomized polynomial-time algorithm, which solves the identification problem for tree-shaped SCMs. For every structural parameter, our algorithms decides whether it is generically identifiable, generically 2-identifiable, or generically unidentifiable. (No other cases can occur.) In the first two cases, it provides one or two fractional affine square root terms of polynomials (FASTPs) for the corresponding parameter, respectively. In particular, our algorithm is not only polynomial time, but also complete for for tree-shaped SCMs.